Naoya SODA, Hiroshi ADACHI
Ibaraki University
Iron loss analysis of EI-shaped iron-core inductor
considering vector magnetic property
Abstract. We analyze the EI-shaped iron-core inductors with air-gaps by using 2-D finite element analysis considering vector magnetic property. We
can investigate the iron loss distribution in the cores affected by the length of the air-gaps. As a result, we can show the difference of local iron loss
distribution in the inductor cores when the length of the air-gaps is changed. Moreover, it is shown that the iron loss distribution differs from the
maximum flux density distribution.
Streszczenie. Przeanalizowano rdzeń typu EI metopdą elementu skończonego z uwzględnieniem właściwości wektorowych. Zbadano wpływ
wielkości szczeliny. Stwierdzono różny rozkład lokalnej wartości strat zależny od wielkości szczeliny. Ten rozkład styrat zależy też od wielkości
indukcji. (Analiza strat w rdzeniu typu EI z uwzględnieniem właściwości wektorowych)
Keywords: finite element analysis, inductor, iron loss, vector magnetic property.
Słowa kluczowe: analiza pola magnetycznego, straty.
Introduction
The important problems in inductors are high efficiency
and saving energy. The two main types of inductor are aircore inductors and iron-core inductors. When high
inductance is necessary, the EI-shaped iron-core inductor
with air-gaps as shown in Fig.1 is generally used. In the
case of the EI-shaped iron-core inductor, the inductance is
often changed by an adjustment of the length of the airgaps. Therefore, from the point of view of saving energy, it
is very important to investigate the iron loss distribution in
the cores when the length of the air-gaps is changed. In
order to investigate the iron loss distribution in the cores,
there are some methods to measure directly the iron loss,
for example, by using the needle probe method [1].
However, when the inductor cores are very small as shown
in Fig. 1, it is impossible to measure the iron loss
distribution in the cores by using the method above.
On the other hand, because the E-core is punched to Eshape from one grain-oriented silicon steel sheet, the
magnetic properties in the core are strongly affected by the
magnetic anisotropy of the steel sheet and the field strength
vector H is not parallel to the flux density vector B. In this
case, an analysis using only the magnetic properties of the
rolling direction and the transverse direction is not able to
consider correctly the phase difference between the vectors
B and H. In addition, because the fringing flux occurs at the
air-gaps of the iron-core inductors, the alternating flux
inclined from the rolling direction and the elliptic rotating flux
occur in the cores. Therefore, it is very effective to
investigate the inductor cores by the magnetic field analysis
considering the two-dimensional (2-D) vector magnetic
property [2, 3].
In this paper, we analyze the EI-shaped iron-core
inductors with air-gaps by using 2-D finite element analysis
with the E&S model [3]. The E&S model can express the
vector magnetic property of both the alternating and the
rotating magnetic flux conditions by using the vector
relationship between the vectors B and H during one period.
As a result, we can investigate the iron loss distribution in
the cores when the length of the air-gaps is changed.
Moreover, we construct the inductors, and measure the
voltage-current characteristics in order to compare with the
analysis results.
Magnetic field analysis
We have carried out the 2-D finite element analysis with
the E&S model, which is expressed as follows:
(1)

 H x  ν xr B x  ν xi

 H y  ν yr B y  ν yi

B x

B y

Fig.1. EI-shaped iron-core Inductor
where xr and yr are named as the magnetic reluctivity
coefficient, xi and yi are named as the magnetic hysteresis
coefficient,  is in a range from 0 to 2. The xr, xi, yr and
yi are expressed as follows:
(2)
2

 B 
 B 
 xr  k xr1  k xr 2 B x2  k xr 3 B x  x   k xr 4  x 

  
  
2

 xi  k xi1  k xi 2 B x2  k xi3 B x  B x   k xi 4  B x 

  
  

2

  k  k B 2  k B  B y   k  B y 
yr1
yr 2 y
yr 3 y 
 yr 4   
 yr
  



2

 B 
 B 
 yi  k yi1  k yi 2 B y2  k yi3 B y  y   k yi 4  y 


  

  



where the coefficients kxrn, kxin, kyrn and kyin (n=1, 2, 3, 4) are
calculated from the measured data by using the 2-D vector
magnetic measurement [3, 4].
From the Maxwell's equations and the E&S model (1),
the governing equation in the 2-D quasi-static magnetic field
can be written as follows:
PRZEGLĄD ELEKTROTECHNICZNY (Electrical Review), ISSN 0033-2097, R. 87 No 9b/2011
81
(3)
 
A   
A 

 yr
   xr
x 
x  y 
y 

 
A   
A 
   J 0
   xi
  yi
x  y 
y 
 x 


where A(=Az) is the magnetic vector potential, J0 is the
exciting current density. Additionally, we assume the eddy
current loss is included in the vector magnetic property.
Because the components of reluctivity coefficients xr, xi,
yr and yi have nonlinearity, it is necessary to calculate
iteratively until those values become constant. We can carry
out the nonlinear magnetic field analysis considering the
vector magnetic property.
(6)
N es


Ptotal  2   D p  N os   Pti  S i  [W]


i 1


where Dp is the thickness of the steel sheet, Nos is the
number of laminations of the steel sheet, Nes is the number
of finite elements in the cores, Pti is the iron loss of a finite
element, and Si is the area of a finite element. In addition,
the total iron loss is doubled because the analytical area is
a half of the inductor as shown in Fig. 2.
Measurement results
In this paper, it is difficult to measure directly the iron
loss distribution in the cores because the inductors are very
small as shown in Fig. 1. Therefore, in order to compare
with the analysis results, we construct the inductors and
measure the voltage-current characteristics. We construct
four kinds of inductors with different air-gap length. The
inductors are one inductor without air-gap, and three
inductors with air-gap length of 0.01, 0.025, and 0.05mm.
The air-gap length is adjusted by inserting a thin insulator
sheet with the same thickness as the gap length in the airgap. In the case without air-gap, no insulator sheet is
inserted in the air-gap. The cores of inductors consist of the
grain-oriented silicon steel sheet (RG11) of 0.35mm
thickness, and the total number of laminations is 32. The
number of coil turns is 72. The frequency of power supply is
50Hz. The values of voltage and current are indicated by
RMS.
Fig.2. Analysis model of EI-shaped iron-core inductor
Fig. 2 shows the analysis model of EI-shaped iron-core
inductor. The arrows in the cores indicate rolling direction. A
half of the region is analyzed because the inductor model
has symmetry. The lengths of three air-gaps between Ecore and I-core are equal as shown in Fig. 2.
In this analysis, we assume that the current waveform is
a perfect sine waveform. Therefore, the terminal voltage of
inductor is calculated from the magnetic vector potential A
as follows:
(4)
J S
A
V 2 
ds  Rc 0 c [V]
t
Nc

C
f
where Cf is the contour along the winding in finite element
region, Rc is the resistance of the winding coil, Sc is the
cross-sectional area of the winding coil, Nc is the number of
turns.
Next, we calculate distribution of the iron loss in the
cores. The iron loss of a finite element can be calculated
directly from the vectors B and H of analysis results by the
following equation:
(5)
dB y 
dB
1 T
Hx x  Hy
dt [ W kg ]
Pt 


0
dt
dt 
T 
where  is the core material density and T is the period of
the exciting waveform. Therefore, the total iron loss of the
inductor can be calculated by the following equation:
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Fig.3. Measured voltage-current characteristics
Fig. 3 shows the measured voltage-current
characteristics of the inductors without air-gap, and with airgap length of 0.01, 0.025, and 0.05mm. We construct four
inductors for each gap length and average the voltagecurrent characteristics. As shown in Fig. 3, as taking air-gap
length large, the characteristics exhibit linearity because
magnetization saturation in the cores becomes gradually
hard. Moreover, when terminal voltage exceeds 2V, the
voltage is saturated under the influence of magnetization
saturation.
Table 1. The relative errors between the measured voltages and
the analyzed voltages for the same current values
0.2A
0.4A
0.6A
0.8A
1.0A
without air-gap
(0.001mm)
24.5%
2.63%
3.96%
0.39%
0.92%
0.01mm
62.1%
9.22%
5.92%
5.03%
3.65%
0.025mm
58.2%
22.1%
4.68%
3.25%
4.49%
0.05mm
36.6%
18.0%
7.69%
2.67%
5.48%
PRZEGLĄD ELEKTROTECHNICZNY (Electrical Review), ISSN 0033-2097, R. 87 No 9b/2011
Analysis results and discussion
We analyze the EI-shaped iron-core inductor model
shown in Fig. 2. In this analysis, the core material is
assumed as the grain-oriented silicon steel sheet (35G155)
that has the characteristics similar to RG11.
An imperceptible air-gap remains between E-core and Icore even if we construct the inductor without air-gap.
Therefore, in this analysis, we analyze the inductor with airgap length of 0.001mm instead of analyzing the inductor
without air-gap.
Fig.4. Analyzed voltage-current characteristics
Fig. 4 shows the voltage-current characteristics
calculated by (4) from analyzed results, and Table 1 shows
the relative errors between the measured voltages shown in
Fig. 3 and the analyzed voltages shown in Fig. 4 for the
same current values. In this analysis, we used the vector
magnetic property that is more than 0.4T. Therefore, we
suppose that the large calculation errors occur at 0.2A
because the reluctivity data less than 0.4T are calculated by
extrapolation. As shown in Fig. 4 and Table 1, analyzed
results except 0.2A are similar to measured ones.
Fig.6. Iron loss distribution in the cores under the current 1.0A
Figs. 5 and 6 show the maximum flux density
distribution and the iron loss distribution in the cores under
the current 1.0A, respectively.
Fig.7. Distribution of the loci of the vectors B and H near the air-gap
(air-gap length=0.01mm, current=1.0A)
Fig.5. Maximum flux density distribution in the cores under the
current 1.0A
Though the maximum flux density is large at only four
corners in the core, the iron loss distribution is quite
different from the maximum flux density distribution. Fig. 6
PRZEGLĄD ELEKTROTECHNICZNY (Electrical Review), ISSN 0033-2097, R. 87 No 9b/2011
83
represents that it is impossible to calculate an iron loss
distribution from a maximum flux density distribution. As
shown in Fig. 6, In the E-cores, large iron loss occurs at the
area magnetized perpendicularly to the rolling direction.
Since the magnetization saturation occurs from the inner
side of corner, the iron loss becomes especially large there.
In the I-cores, because magnetic flux bends near air-gaps
when magnetic flux flows into the I-core from the E-core,
large iron loss occurs near air-gaps. Fig. 7 shows
distribution of the loci of the vectors B and H near the airgap between E-core and I-core. In Fig. 7, the air-gap length
is 0.01mm and the current is 1.0A. As shown in Fig. 7 (a),
the alternating flux inclined from the rolling direction and the
elliptic rotating flux occur in the I-core. Moreover, because
many rotating fields occur at the corner part in the I-core as
shown in Fig. 7 (b), the iron loss becomes large there as
shown in Fig. 6 (b).
Fig.9. Iron loss distribution in the cores under the comparatively
similar voltage
Fig.8. Relationship of total iron loss of the inductors to the voltage
Next, we calculate the total iron loss of the inductors by
(6) from the local iron loss distribution as shown in Fig. 6.
Fig. 8 shows the relationship of the total iron loss to the
terminal voltage. In this analysis, the terminal voltage
represents the average magnetic flux in the inductor. When
the air-gap length is enlarged, the average magnetic flux in
the inductor decreases and the total iron loss also
decreases. Therefore, as shown in Fig. 8, the values of total
iron loss of the inductors with different air-gap length are
almost on the same curve. The total iron loss is the average
value and is hard to reflect the variation of local iron loss.
We compare the local iron loss distribution under the
comparatively similar voltage surrounded with the circle
dashed line in Fig. 8.
Fig. 9 shows the iron loss distribution of the inductor
with different air-gap length under the comparatively similar
voltage. As shown in Figs. 9 (a) and (b), when the air-gap
length is small, the iron loss is large at the inner side of
corner in the E-core. On the other hand, as shown in Fig. 9
(d), when the air-gap length is large, large value of iron loss
is distributed widely near the air-gap in the I-core. As a
result, we can investigate the difference of local iron loss
distribution in the inductor cores when the length of the airgaps is changed.
Conclusion
In this paper, we analyze the EI-shaped iron-core
inductors with air-gaps by using 2-D finite element analysis
with the E&S model and obtain the iron loss distribution in
the cores when the length of the air-gaps is changed.
It is shown that the iron loss distribution differs from the
maximum flux density distribution. Therefore, it is very
important to obtain the iron loss distribution directly by using
magnetic field analysis with the E&S model considering
vector magnetic property.
Moreover, we can show the difference of local iron loss
distribution in the inductor cores when the length of the airgaps is changed.
REFERENCES
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[2] Zhang Y., Eum Y. H., Xie D., Koh C. S., An Improved
Engineering Model of Vector Magnetic Properties of GrainOriented Electrical Steels, IEEE Trans. on Magn., 44(2008),
No. 11, 3181-3184
[3] Soda N., Enokizono M., Improvement of T-Joint Part
Constructions in Three-Phase Transformer Cores by Using
Direct Loss Analysis with E&S Model, IEEE Trans. on Magn.,
36(2000), No. 4, 1285-1288
[4] Soda N., Enokizono M., E&S hysteresis model for twodimensional magnetic properties, Journal of Magnetism and
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Authors: dr. Naoya SODA, Department of Electrical and Electronic
Engineering, College of Engineering, Ibaraki University, Hitachi,
Ibaraki, 316-8511 Japan, E-mail: soda@mx.ibaraki.ac.jp
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PRZEGLĄD ELEKTROTECHNICZNY (Electrical Review), ISSN 0033-2097, R. 87 No 9b/2011